Lecture 1: linear optimization: introduction. What is optimization? What is a cost function? Linear or affine cost functions.

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1 Lecture : linear optimization: introduction What is optimization? Definition of cost / objective function Example of cost functions, affine functions, linear functions Definition of constraints Example of constraints, linear constraints Linear programs General form of a linear program Sigma notation Extended example : the transportation problem Google maps Extended example : the shortest path problem or Minimize a cost or objective function (for ex. cost of production) imize a cost or objective function (for ex. profit) with respect to constraints - Employee cannot work more than x hours a day - Only three people can use the same machine at a time - The pipeline s maximal fuel throughput is y i.e. find a solution that is optimal within limits given What is a cost function? Linear or affine cost functions cost cost approximation pond size Example: cost of a mile as a function of the distance For some application, some cost functions look almost like lines, i.e. are linear or affine. Example here: cost of building a dam as a function of the size of the pond [ [ Linar functions ffine functions Price of gas ($): 0 0 gallon = $ 0 gallon = $. gallons = $.0 Price of renting a U-haul ($): $. 0 gallon 0 X 0 (mile)

2 Linear or affine cost functions: formal definition Minimizing affine or linear function is the same Minimizing the affine cost function Minimizing a function f(x) f f(x) is the same as minimizing the linear cost function more general expression of the cost function: x Minimizing affine or linear function is the same Example: cost of building a wall Minimizing a function f(x) or f(x)+c is the same f f(x) f(x)+c C<0 Cost of a pound of cement ($ per lb) Cost of a feet of steel beam ($ per ft) Weight of cement (lb) Length of steel beam (ft) Total cost ($) Note that none of the variables above has the same unit! Minimum obtained at the same x x Note however that and and have the same unit What is a constraint? constraint is a condition on variables which restricts the values they can take Your maximal budget for cement is Summary Your optimization program incorporating all your constraints can be formulated as follows. Your minimal budget for steel is You want to spend twice as much for steel as for cement You want to spend a given minimum amount for the wall

3 Constraints in the form of equalities (I) Sometimes, constraints are given in the form of equalities Constraints in the form of equalities (II) So you could rewrite the program in the following form: Example: you want to spend exactly twice as much for steel as for cement: This is exactly the same as and One can thus assume that all constraints are always given in the form of inequalities. General form for a linear program So you could rewrite the program in the following form: Sigma notation So you could rewrite the program in the following form: Example: the transportation problem (I) s farm produces tons of apples per day Ron s farm produces tons of apples per day s factory needs ton of apples per day ob s factory needs tons of apples per day George owns both farms and factories. He is paying the cost of shipping all the apples from the farms to the factories. The shipping costs for George are: : 000$ per ton Ron : 0$ per ton ob: 0$ per ton Ron ob: 0$ per ton Example: the transportation problem (II) George pays for the shipping 000$ 0$ 0$ Ron 0$ ob What is the best way to ship the apples?

4 Example: the transportation problem (III) Example: the transportation problem (IV) Ron 000$ 0$ 0$ 0$ ob Ron 000$ 0$ 0$ 0$ ob General form of the transportation problem Please, be lazy, do not write pages of equations Use summations, they leave you more time to go to the movies 000$ 0$ Ron 0$ 0$ ob Example: a «small» network (air traffic control) Example: a «large» network (the internet) [Robelin, Sun, ayen, tech. rep., 00] [

5 0 : length of the shortest path : length of the shortest path 0 0 Define For every (i,j) on the shortest path For every (i,j) not on the shortest path : length of the shortest path : choice of path 0 Define Define For every (i,j) on the shortest path For every (i,j) on the shortest path For every (i,j) not on the shortest path For every (i,j) not on the shortest path

6 Define a graph (road network) Define a graph (road network) Call the cost to go from i to j (for exmple fuel burned) For example is the cost to go from node to node Take if decides to go through link (i,j), zero otherwise For example if decides to use route (,) ll other are zero Total length of this path: Total length: Minimize: Total length Cost of arc (i,j) if link (i,j) chosen 0 otherwise ll links arriving at node i ll links leaving node i set of nodes j with direct connections to node i set of nodes j with direct connections to node i

7 Starting from, can only take one path rriving at, one can only take one path set of nodes j with direct connections to node set of nodes j with direct connections to node Example: a «small» network (air traffic control) set of nodes j with direct connections to node i [Robelin, Sun, ayen, tech. rep., 00] Example: a «large» network (the internet) [